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Polynomial Optimization, Certificates of Positivity, and Christoffel Function

In: Polynomial Optimization, Moments, and Applications

Author

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  • Jean B. Lasserre

    (LAAS-CNRS and Institute of Mathematics)

Abstract

We briefly recall basics of the Moment-SOS hierarchy in polynomial optimization and the Christoffel-Darboux kernel (and the Christoffel function (CF)) in theory of approximation and orthogonal polynomials. We then (i) show a strong link between the CF and the SOS-based positive certificate at the core of the Moment-SOS hierarchy, and (ii) describe how the CD-kernel provides a simple interpretation of the SOS-hierarchy of lower bounds as searching for some signed polynomial density (while the SOS-hierarchy of upper bounds is searching for a positive (SOS) density). This link between the CF and positive certificates, in turn allows us (i) to establish a disintegration property of the CF much like for measures, and (ii) for certain sets, to relate the CF of their equilibrium measure with a certificate of positivity on the set, for constant polynomials.

Suggested Citation

  • Jean B. Lasserre, 2023. "Polynomial Optimization, Certificates of Positivity, and Christoffel Function," Springer Optimization and Its Applications, in: Michal Kočvara & Bernard Mourrain & Cordian Riener (ed.), Polynomial Optimization, Moments, and Applications, pages 1-22, Springer.
    • Handle: RePEc:spr:spochp:978-3-031-38659-6_1
    DOI: 10.1007/978-3-031-38659-6_1
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